Download Curriculum Cluster 2

Geometry
Possible Scope and Sequence
Curriculum Cluster 2:
Parallel and Perpendicular Lines
a.1 Builds on knowledge of number, operation, and quantitative reasoning; patterns, and algebraic thinking, geometry, measurement, and probability and statistics, solve
meaningful problems by representing and transforming figures and analyzing relationships.
a.6 Makes connections, uses multiple representations, technology, applications and modeling, and numerical fluency in problem solving contexts
G.1 The student understands the structure of, and relationships within, an axiomatic system.
G.2 The student analyzes geometric relationships in order to make and verify conjectures.
G.3 The student applies logical reasoning to justify and prove mathematical statements .
G.4 The student uses a variety of representations to describe geometric relationships and solve problems.
G.7 The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly.
G.9 The student analyzes properties and describes relationships in geometric figures.
Possible Resources
TEKS
G.1.A
Develop awareness of the structure
of a mathematical system,
connecting definitions,
postulates, logical reasoning, and
theorems.
G.3.C
Use logical reasoning to prove
statements are true and find counter
examples to disprove
statements t are false.
Instructional Scope
Logic and Truth Tables:
Introduction to Symbolic Logic
When you answer true –false questions on a
test, the basic principle of logic is applied. You
know that there is only one correct answer,
either true or false. The truth or falsity of a
statement is called its truth value.
Instructional
Resources
Assessment
Resources
Supplemental
Resources
Holt: Chapter 2 Introduction to
Symbolic Logic,
pp.128-129
Holt Lesson Tutorial
Videos: CD-ROM
Chapter 2
A convenient method for organizing and listing
all possible combinations of truth values for a
statement or compound statements is called a
truth table.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 1 of 36
Geometry
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TEKS
G.4.A
Select an appropriate representation
(concrete, pictorial, graphical, verbal,
or symbolic) in order to solve
problems.
Instructional Scope
Instructional
Resources
Assessment
Resources
Supplemental
Resources
Compound Statements:
Conjunction – p ˅ q ( read “ p and q”) is true
when all of its parts are true.
Union symbol - ˅
Disjunction – p ˄ q (read “ p or q”) is true if
any one of its parts are true.
Intersection symbol - ˄
Truth tables can be constructed for more
complex compound statements involving
negations ( ̴ p – read “not p”), conjunctions,
and disjunctions.
Example:
p
T
T
F
F
q
T
F
T
F
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
̴ p
F
F
T
T
̴ p˄q
F
F
T
F
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 2 of 36
Geometry
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TEKS
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Resources
Assessment
Resources
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Resources
Ask students to construct and complete truth
tables to determine the truth value of logical
and given statements for p and q.
G.1.A
Develop an awareness of the
structure of a mathematical system,
connecting definitions, postulates
and theorems.
G.1.B
Recognize the historical
development of geometric systems
and know mathematics is developed
for a variety of purposes.
G.1.C
Compare and contrast the structures
and implications of Euclidean and
non-Euclidean
geometries.
Non-Euclidean Geometries
Euclid was the founder of plane geometry.
Therefore the geometry we use with planes or
about planes is called Euclidean Geometry.
Euclid used postulates and deductive reasoning
.
There are other geometries called NonEuclidean geometries. Hyperbolic geometry is
a non-Euclidean geometry. It is known for
having constant curvature. This geometry
satisfies all of Euclid’s postulates except the
parallel postulate. Rather than having one line
through a point parallel to a given line,
hyperbolic geometry uses many lines parallel
to the given line.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Holt : Chapter 10Spherical Geometry,
pp. 726-729
High School
Geometry:
Supporting TEKS
and TAKS
(TEXTEAMS).
“Bayou City
Geometry,”
Activities 6-9, pp.
20-26.
High School
Geometry:
Supporting TEKS
and TAKS
(TEXTEAMS) “Bayou
City Geometry”
Reflect and Apply, p.
27.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 3 of 36
Geometry
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TEKS
G.7.A
Use one and two dimensional
coordinate systems to represent
points, lines, rays, line segments, and
systems to represent points, lines,
rays, line segments, and figures.
G.9.D
Analyze the characteristics of
polyhedra and other 3-D figures
and their component parts based on
explorations and models.
G.7.C
Use formulas involving length,
slope, and midpoint.
and midpoint.
Instructional Scope
Euclidean Geometries are spherical geometry
and taxi cab geometry. In spherical geometry
the lines are like latitude and longitude of the
sphere.
In taxi cab geometry, distance is used.
Although this geometry does use some
planes, the distance is not “as the crow flies”.
In taxi cab geometry, the distance between
two points is how a taxi would travel to get
from one point to another. There are usually
many ways to get there, but must travel in
horizontal or vertical directions.
Taxi Cab Geometry
Find the distance from A to B
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Instructional
Resources
High School
Geometry:
Supporting TEKS
and TAKS
(TEXTEAMS).
“Spherical
Geometry,” pp.
223-228.
Assessment
Resources
Supplemental
Resources
Engaging
Mathematics: TEKS
Based Activities
Geometry
Non-Euclidean
Geometry, pp.24-25
Geometry Clarifying
Activities
http://www.tenet.ed
u/teks/math/clarifyin
g/cageob1.html
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 4 of 36
Geometry
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TEKS
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Resources
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Resources
Red route:
7 blocks left
2 blocks up
Total distance= 9 blocks
Green route:
1 block up
5 blocks left
1 block up
2 blocks left
Total distance= 9 blocks
The red and green routes are only 2 of
the possible routes from A to B.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 5 of 36
Geometry
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TEKS
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Instructional
Resources
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Resources
Supplemental
Resources
 Compare and contrast Euclidean and nonEuclidean (spherical, hyperbolic, and taxicab)
geometries in order to illustrate the importance of
precise definitions and application of postulates.
For example, ask students to brainstorm about
triangles. Sample answers might be: three sides,
three angles, and all angles add up to 180o . Ask
them to brainstorm about lines. Sample answers
might be: they go on for forever; it only takes two
points to make a line.
Give students a tennis ball and three rubber
bands (one set per group). Explain the definition
of a line on a sphere is a great circle—which is a
line on the surface of a sphere formed by the
intersection of a plane through the center of the
sphere. Have them place one rubber band
around the tennis ball letting it represent a line
(like an equator).
Ask students what they know about
perpendicular lines. See if they can place
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 6 of 36
Geometry
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Resources
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Resources
another rubber band on the tennis ball that is
perpendicular to the first one.
Have them find the 90o angles that are formed.
By placing a third rubber band on the tennis ball
perpendicular to the second rubber band, the
students will form a triangle, with three sides that
have three 90o angles for a total of 270o .
Use questioning techniques to help students
draw conclusions.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 7 of 36
Geometry
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TEKS
G.1.A
Develop an awareness of the
structure of a mathematical system,
connecting definitions, postulates
and theorems.
G.5.B
Use numeric and geometric patterns
to make generalizations about
geometric properties,
including properties of polygons,
ratios in similar figures and solids,
and angle relationships in
polygons and circles.
G.9.A
Formulate and test conjectures about
the properties of parallel and
perpendicular lines based on
explorations and concrete models.
Instructional Scope
Transversals and Special Pairs of
Angles
Key Vocabulary:
transversal, exterior angles,
interior angles, corresponding angles,
alternate interior angles, alternate exterior
angles, consecutive interior angles
A line that intersects two other lines at different
points is called a transversal.
Two lines and a transversal form eight angles.
Some pairs of these angles have special names.
Instructional
Resources
Holt Geometry –
Chapter 3
pp. 146-211
Resources for All
Learners
Holt Resource
Book:
Chapter 3
Developing
LearnersPractice A,
Reteach,
Questioning
Strategies, Modified
Chapter 3
Resources
On-Level
LearnersPractice B, Multiple
Representations,
Cognitive Strategies
In the figure, t is the transversal that intersects
lines m and n.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Advanced
LearnersPractice C,
Challenge,
Critical Thinking
Assessment
Resources
Instructional Fair
Geometry
8764(IF0280)
pp.32-35.
Holt: Chapter 3
Test & Practice
Generator – OneStop Planner
Chapter Test
(Levels A, B, C)
Holt Assessment
Resources,
pp. 47-58
Supplemental
Resources
Holt 3.1 -3.6
Power-Point
Presentations
CD-ROM
Holt 3.1-3.6
Lesson Tutorial
Videos
Holt: 3.1-3.6 One
Stop Planner,
CD-ROM, Disc 1
Holt Texas Lab
Manual:
Geometry Lab
Chapter 3
www.kutasoftware.
com/freeige.html
Parallel Lines and
Transversals
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 8 of 36
Geometry
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TEKS
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Resources
Assessment
Resources
Supplemental
Resources
Holt
Exterior angles are angles on the outside of the
two lines. In the figure, angles 1, 2, 7, and 8 are
exterior angles.
Technology Lab:3.2
Explore Parallel and
Perpendicular
Lines, p. 154
online.dpsk12.org./
math-resources /
Interior angles are angles that are inside the
two lines. In the figure, angles 3, 4, 5, and 6 are
interior angles.
Corresponding angles are angles in the same
positions on the two lines.
In the figure, < 1 and < 5 are corresponding
angles.
The other pairs of corresponding angles are
< 2 and < 6, <4 and < 8, and <3 and <7.
Alternate exterior angles are angles outside
the two lines and on opposite sides of the
transversal.
< 1 and < 7 are alternate exterior angles.
< 2 and < 8 are alternate exterior angles.
Possible Scope and Sequence, Geometry
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Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 9 of 36
Geometry
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TEKS
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Resources
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Resources
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Resources
Consecutive interior angles are angles inside
the two lines and on the same side of the
transversal.
< 3 and < 6 are consecutive interior angles.
< 4 and < 5 are consecutive interior angles.
Example:
Identify the lines and transversal that form the
pair of angles.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 10 of 36
Geometry
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Resources
< 9 and < 13
< 9 is formed by lines q and n.
< 13 is formed by lines q and l.
Therefore, the two lines are n and l, and the
transversal is q.
Example:
Identify which pair of special angles
< 3 and < 9
are.
< 3 and < 9 are inside lines p and q, and on
opposite sides of transversal n.
Therefore, < 3 and < 9 are alternate interior
angles.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 11 of 36
Geometry
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TEKS
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Resources
Assessment
Resources
Supplemental
Resources
Parallel Lines and Transversals:
Visual Activity:
Have students use lined paper to draw two
parallel lines and a transversal that is not
perpendicular to the lines. Instruct students to
shade the acute angles one color and the obtuse
angles another color. Let students use a
protractor to see that all the angles shaded the
same color are congruent and that pairs of
angles shaded different colors are
supplementary/
Holt Manipulative
Kit:
protractors
patty paper
Kinesthetic Activity:
Have students draw a pair of parallel lines and
a transversal on patty paper. Tear the paper
between the parallel lines and overlay the two
parts to show that the angles are congruent.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 12 of 36
Geometry
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TEKS
G.1.A
Develop an awareness of the
structure of a mathematical system,
connecting definitions, postulates
and theorems.
Instructional Scope
Instructional
Resources
ANGLES FORMED BY PARALLEL LINES
Holt 3.3- Proving
Lines Parallel,
pp. 162-169
KEY VOCABULARY:
Parallel lines, skew lines
Two lines in the same plane that do not intersect
are parallel lines. Two lines that are not in the
same plane, but do not intersect , are skew lines.
G.2.A
Use constructions to explore
attributes of geometric figures and
make conjectures about geometric
relationships.
When using parallel lines, the symbol on a
figure is a larger bolder arrow, or the larger
arrows in a different color.
In writing, the symbol is ||.
Assessment
Resources
Supplemental
Resources
Engaging
Mathematics:
TEKS Based
Activities
Geometry
Angle Relationships,
Activity 2, p. 36
Angle Puzzler, p.37
Angle Relationships,
p. 38
Parallel Universe,
p. 39
G.3.C
Use logical reasoning to prove
statements are true and find
counterexamples to disprove
statements that are false.
In the figure, m ||n.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 13 of 36
Geometry
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TEKS
Instructional Scope
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Resources
Assessment
Resources
Supplemental
Resources
We know that two lines intersected by a
transversal form special pairs of angles. When
these lines are parallel, the special pairs of
angles have special relationships.
Corresponding Angles Postulate
If two lines are parallel and cut by a
transversal, then each pair of corresponding
angles are congruent.
1
2
4
 3




Geometry Lab:3.3
Constructing
Parallel Lines,
p.170
5
6
8
7
Alternate Interior Angles Theorem
If two lines are parallel and cut by a
transversal, then each pair of alternate interior
angles are congruent.
3  5
 4  6
Possible Scope and Sequence, Geometry
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Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 14 of 36
Geometry
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TEKS
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Resources
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Resources
Supplemental
Resources
Alternate Exterior Angles Theorem
If two lines are parallel and cut by a
transversal, then each pair of
alternate exterior angles are congruent.
1  7
2  8
Consecutive Interior Angles Theorem
If two lines are parallel and cut by a
transversal, then each pair of
consecutive interior angles are supplementary.
 4 and  5 are supplementary.
 3 and  6 are supplementary.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 15 of 36
Geometry
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TEKS
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Resources
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Resources
Example:
Given m ||n.
If m  1 = 120°, find m  8.
The m  5 = 120 because  1 and  5 are
corresponding angles.
m  5 + m  8 = 180°, since they
form a linear pair.
120 + m  8 = 180
m  8 = 60°.
Example:
Using the figure above,
if m  4 =80°, find the m  7.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 16 of 36
Geometry
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TEKS
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Resources
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Resources
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Resources
 4 and  5 are supplementary
because they are consecutive
interior angles.
m  4 + m  5 = 180°
80 + m  5 = 180
m  5 = 100°.
Therefore, m  7 = 100° because
 5 and  7 are vertical angles,
and vertical angles are congruent.
Example:
Using the figure above,
if m  3 =5x – 9 and m  5 = 3x + 3, find the
value of x.
 3 and  5 are alternate interior angles, so
they are congruent.
m 3 = m5
5x – 9 = 3x + 3
2x – 9 = 3
2x = 12
x=6
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 17 of 36
Geometry
Possible Scope and Sequence
Possible Resources
TEKS
Instructional Scope
Instructional
Resources
Assessment
Resources
Supplemental
Resources
Identifying or Proving Parallel Lines
G.9.A
Formulate and test conjectures about
the properties of parallel and
perpendicular lines based on
explorations and concrete models.
Sometimes we have to determine if lines are
parallel. To do this, we need to use the angle
relationships and the converses of the Parallel
Lines theorems and postulate.
Corresponding Angles Converse
If two lines are cut by a transversal and
corresponding angles are congruent, then the
lines are parallel.
Alternate Interior Angles Converse
If two lines are cut by a transversal and alternate
interior angles are congruent, then the lines are
parallel.
Alternate Exterior Angles Converse
If two lines are cut by a transversal and alternate
exterior angles are congruent, then the lines are
parallel.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 18 of 36
Geometry
Possible Scope and Sequence
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TEKS
Instructional Scope
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Resources
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Resources
Supplemental
Resources
Consecutive Interior Angles Converse
If two lines are cut by a transversal and
consecutive interior angles are
supplementary, then the lines are
parallel.
Example:
Use the given information to determine
if lines m and n are parallel.
p
1
2
3
m
4
5
6
n
7 8
1  5
m || n
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 19 of 36
Geometry
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TEKS
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Resources
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Resources
The angles are corresponding angles, and by
the Corresponding Angles Converse, the lines
are parallel.
3 6
m || n
The angles are alternate interior angles, and by
the Alternate Interior Angles Converse, the lines
are parallel.
 2 supplementary to  6
m is not parallel to n
The angles are corresponding. In order for the
lines to be parallel, corresponding angles must
be congruent, not supplementary.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 20 of 36
Geometry
Possible Scope and Sequence
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TEKS
Instructional Scope
Instructional
Resources
Assessment
Resources
Supplemental
Resources
Perpendicular Lines
G.2.A
Use constructions to explore
attributes of geometric figures and
make conjectures about geometric
relationships.
G.3.E
Use deductive reasoning to prove a
statement.
Key vocabulary: Perpendicular Lines
Two lines are perpendicular if they intersect to
form right angles.
Rays and segments are parts of lines, and can
also be perpendicular.
Holt 3.4 –
Perpendicular
Lines, pp. 172178
AB  CD
G.9.A
Formulate and test conjectures about
properties of parallel and
perpendicular lines based on
explorations and concrete models.
C
A
B
D
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 21 of 36
Geometry
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TEKS
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Instructional
Resources
Assessment
Resources
Supplemental
Resources
XP  XC
P
X
C
Holt Geometry
Lab:3.4Constructing
Perpendicular
Lines, p. 179
Example:
If m  XQY = 2x and
m  YQZ = 4x + 6,
find the value of x.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 22 of 36
Geometry
Possible Scope and Sequence
Possible Resources
TEKS
Instructional Scope
Instructional
Resources
Assessment
Resources
Supplemental
Resources
Since the angles have a sum
of 90 degrees,
2x + 4x + 6 = 90
6x + 6 = 90
6x = 84
X = 14
Engaging
Mathematics:
TEKS Based
Activities
Geometry
(Slope of a Line)
Time to Go Home,
p. 31
Slopes of Lines
Key Vocabulary: slope
G.7.A
Use one- and two-dimensional
coordinate system to represent
points, lines, rays, line segments, and
figures.
The slope of a line is the ratio of vertical change
to horizontal change. The only exception to this
is the vertical line, which has an undefined
slope.
The slope m = change in y = rise
change in x run
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Holt 3.5 – Slopes
of LinesH
pp. 182-187
One and TwoDimensional
Coordinate System,
Activity 3, p. 32
Which Line is It,
Anyway?
p. 33
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 23 of 36
Geometry
Possible Scope and Sequence
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TEKS
G.7.B
Use slopes and equations of lines to
investigate geometric relationships.
Instructional Scope
Instructional
Resources
Assessment
Resources
Supplemental
Resources
or m = y2 – y1
x2 – x1
Example:
Find the slope of the line.
G.7.C
Derive and use formulas involving
length, slope, and
midpoint.
Determine the points on the graph.
The points are ( 1,1) and ( -1, -2).
m = -2 - 1
-1 – 1
m = -3
-2
m=3
2
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 24 of 36
Geometry
Possible Scope and Sequence
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TEKS
Instructional Scope
Notice the line rises left to right, and
the slope is positive.
Similarly, when a line has a negative
slope, it will fall left to right.
Instructional
Resources
Assessment
Resources
Supplemental
Resources
www.kutasoftware.
com/freeige.html
Parallel Lines in the
Coordinate Plane
Parallel lines have the same slope.
Perpendicular lines have slopes that are
opposite reciprocals.
Example:
Determine the slope of a line parallel to the line
which contains points ( -5, 9) and ( -4, 7).
Find the slope of the line containing these
points.
m=7-9
-4 – (-5)
m = -2
1
m = -2
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 25 of 36
Geometry
Possible Scope and Sequence
Possible Resources
TEKS
Instructional Scope
Instructional
Resources
Assessment
Resources
Supplemental
Resources
The slope of this line is - 2. Parallel lines have
equal slopes. Therefore, any line parallel to the
line with slope -2 will also have slope of -2.
Example:
Determine the slope of any line perpendicular to
the line which contains the points ( 2, 0) and
( 4, -6).
Find the slope of the line containing the points.
m = -6 – 0
4–2
m = -6
2
m = -3
The slope of this line is -3.
Perpendicular lines have slopes that are opposite
reciprocals. Therefore, any line that is
perpendicular to the line with slope -3 will have
a slope of 1 .
3
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 26 of 36
Geometry
Possible Scope and Sequence
Possible Resources
TEKS
G.7.B
Use slopes and equations of lines to
investigate geometric relationships,
including parallel lines
perpendicular lines, and
special segments of triangles and
other polygons.
Instructional Scope
Instructional
Resources
Assessment
Resources
Supplemental
Resources
Investigating Slopes
A graphing calculator can help in exploring
graphs of parallel and perpendicular lines. To
graph a line on a calculator, enter the equation
of the line in slope-intercept form. The slope
intercept form of the equation is y = mx + b,
where m is the slope and b is the y-intercept.
Example:
y = 2x + 3 has a slope of 2 and crosses the yaxis at (0,3).
Holt Technology
Lab 3.6:
pp. 188-189
Activity:
Have students use the table feature to make a
table of values for y = 3x -4 and y = 3x + 1.
Use the table to find points on each line, and
calculate the slopes to verify that the lines are
parallel. Then make a table of values for
y = 3x – 4 and y = - 1/3 x , find two points,
and calculate the slopes to verify that the lines
are perpendicular.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 27 of 36
Geometry
Possible Scope and Sequence
Possible Resources
TEKS
G.7A
Use one and two-dimensional
coordinate systems to represent
points, lines, rays, line segments, and
figures.
G.7B
Use slopes and equations of lines to
investigate geometric relationships,
including parallel lines,
perpendicular lines, and special
segments of triangles and other
polygons.
G. 7C
Derive and use formulas involving
length, slope, and midpoint.
Instructional Scope
Instructional
Resources
Assessment
Resources
Supplemental
Resources
Writing Equations of Lines
Key Vocabulary: slope- intercept form
Linear equations can be written in several
forms. The most common is the slopeintercept form, y = mx + b, where m is the
slope and b is the y intercept.
y – y1 = m ( x – x1) is called the point- slope
form of the equation. We will need to use this
equation to input our values, then solve the
equation for y to get the slope-intercept
equation.
Holt 3.6 – Lines
in the Coordinate
Plane, pp. 190197
tx.geometryonline
.com
Example:
Write the equation of the line that has slope 2,
and goes through the point ( -3, 4 ).
Using the slope, m= 2, and the point ( -3, 4 ) as
( x1, y1 ), we will write the equation in pointslope form.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 28 of 36
Geometry
Possible Scope and Sequence
Possible Resources
TEKS
Instructional Scope
Instructional
Resources
Assessment
Resources
Supplemental
Resources
y – 4 = 2 ( x – ( -3) )
y – 4 = 2 ( x + 3)
Solving the equation for y,
y – 4 = 2x + 6
y = 2x + 10
Example:
Write the equation of the line that contains the
points ( 0, -6 ) and ( -4 , - 3).
We need to find the slope of the line.
m=
6  (3)
0  (4)
m=
3
4
Use this slope with one of the points.
It does not matter which one.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Geometry
Assessments: The
Charles A. Dana
Center,
Cross Country
Cable, pp. 5-12
Whitebeard’s
Treasure, pp. 13-18
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 29 of 36
Geometry
Possible Scope and Sequence
Possible Resources
TEKS
Instructional Scope
Instructional
Resources
Assessment
Resources
Supplemental
Resources
3
and the
4
point ( 0, -6) in the point- slope equation.
We will use the slope
y – (-6 ) =
y+6=
y=
G.7A
Use one and two-dimensional
coordinate systems to represent
points, lines, rays, line segments, and
figures.
G.7B
Use slopes and equations of lines to
investigate geometric relationships,
including parallel lines,
perpendicular lines, and special
segments of triangles and other
polygons.
3
( x – 0)
4
3
x
4
3
x -6
4
Parallel lines have the same slope.
Perpendicular lines have slopes that
are opposite reciprocals.
Standard – Form of a Linear Equation in
Two Variables
Ax + By = C, where A and B are not both
zero.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 30 of 36
Geometry
Possible Scope and Sequence
Possible Resources
TEKS
G. 7CDerive and use formulas
involving length, slope, and
midpoint.
Instructional Scope
Instructional
Resources
Assessment
Resources
Supplemental
Resources
Write the standard-form of the equation in
slope- intercept form ( solve for y):
y = -A/B x + C/B
m = -A/B
(slope)
and
b = C/B
(y-intercept)
So a quick way to check whether lines have the
same slope is to find –A/B for both lines. If the
slopes are the same, then find C/B to see if the
lines coincide(same slope and same y-intercept).
Example:
Write the equation of the line that goes through
( 3, -4) and is parallel to the line
2x – 3y =6.
Find the slope of the line. To do this, we solve
for y, the slope intercept form of the equation.
2x – 3y = 6
- 3y = - 2x + 6
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 31 of 36
Geometry
Possible Scope and Sequence
Possible Resources
TEKS
Instructional Scope
y=
Instructional
Resources
Assessment
Resources
Supplemental
Resources
2
x -2
3
The slope of the line is
2
.
3
To find the equation of a line that is parallel to
this line, we will use the same slope, and the
given point.
y – ( -4) =
y+4=
y=
2
(x–3)
3
2
x -2
3
2
x -6
3
This is the equation of the line through the given
point, and parallel to the given line.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 32 of 36
Geometry
Possible Scope and Sequence
Possible Resources
TEKS
Instructional Scope
Instructional
Resources
Assessment
Resources
Supplemental
Resources
Example:
Write the equation of the line that contains the
point ( 1, -5) and is perpendicular to the line
with the equation -3x + 4y = 12.
We find the slope of the given line.
-3x + 4y = 12
4y = 3x + 12
3
y= x+3
4
3
. The slope of a line
4
perpendicular to it will be the opposite
3
reciprocal of . The new slope to use will be
4
4
.
3
The slope of this line is
The equation of the line that goes through
4
( 1, -5 ) and has slope
is
3
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 33 of 36
Geometry
Possible Scope and Sequence
Possible Resources
TEKS
Instructional Scope
y – ( -5) =
y+5=
y=
Instructional
Resources
Assessment
Resources
Supplemental
Resources
4
( x – 1)
3
4
4
x +
3
3
4
11
x 3
3
This line is perpendicular to the line whose
equation is – 3x + 4y = 12.
Equations of Vertical and Horizontal Lines
Vertical Line: x = a real number
The slope of a vertical line is undefined
(no slope).
Horizontal Line: y = a a real number
The slope of a horizontal line is zero.
Slope- Intercept Form: y = 0x + b
( horizontal line)
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 34 of 36
Geometry
Possible Scope and Sequence
Possible Resources
TEKS
G.7B
Use slopes and equations of lines to
investigate geometric relationships,
including parallel lines,
perpendicular lines, and special
segments of triangles and other
polygons.
Instructional Scope
Activity:
Ask students to draw a pair of parallel lines, a
pair of intersecting lines, and than write a
system of equations for each pair. Some
students may prefer solving systems of
equations by other methods, such as graphing
or substitution.
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Instructional
Resources
Assessment
Resources
Supplemental
Resources
Holt: Systems of
Equations –
Algebra Review,
pp. 152-153
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 35 of 36
Geometry
Possible Scope and Sequence
Possible Resources
TEKS
Instructional Scope
Possible Scope and Sequence, Geometry
© 2005 Region 4 Education Service Center. All rights reserved.
Instructional
Resources
Assessment
Resources
Supplemental
Resources
Curriculum Cluster 2: Parallel and Perpendicular Lines
Page 36 of 36